Reconstructive Integral Geometry

One hundred years ago (1904) Hermann Minkowski [58] posed a problem: to re 2 construct an even function I on the sphere 8 from knowledge of the integrals MI (C) = fc Ids over big circles C. Paul Funk found an explicit reconstruction formula for I from data of big circle integrals. Johann Radon studied a similar problem for the Eu clidean plane and space. The interest in reconstruction problems like Minkowski Funk's and Radon's has grown tremendously in the last four decades, stimulated by the spectrum of new modalities of image reconstruction. These are X-ray, MRI, gamma and positron radiography, ultrasound, seismic tomography, electron mi croscopy, synthetic radar imaging and others. The physical principles of these methods are very different, however their mathematical models and solution meth ods have very much in common. The umbrella name reconstructive integral geom etryl is used to specify the variety of these problems and methods. The objective of this book is to present in a uniform way the scope of well known and recent results and methods in the reconstructive integral geometry. We do not touch here the problems arising in adaptation of analytic methods to numerical reconstruction algorithms. We refer to the books [61], [62] which are focused on these problems. Various aspects of interplay of integral geometry and differential equations are discussed in Chapters 7 and 8. The results presented here are partially new.

1 Distributions and Fourier Transform. - 1. 1 Introduction. - 1. 2 Distributions and generalized functions. - 1. 3 Tempered distributions. - 1. 4 Homogeneous distributions. - 1. 5 Manifolds and differential forms. - 1. 6 Push down and pull back. - 1. 7 More on the Fourier transform. - 1. 8 Bandlimited functions and interpolation. - 2 Radon Transform. - 2. 1 Properties. - 2. 2 Inversion formulae. - 2. 3 Alternative formulae. - 2. 4 Range conditions. - 2. 5 Frequency analysis. - 2. 6 Radon transform of differential forms. - 3 The Funk Transform. - 3. 1 Factorable mappings. - 3. 2 Spaces of constant curvature. - 3. 3 Inversion of the Funk transform. - 3. 4 Radon s inversion via Funk s inversion. - 3. 5 Unified form. - 3. 6 Funk-Radon transform and wave fronts. - 3. 7 Integral transform of boundary discontinuities. - 3. 8 Nonlinear artifacts. - 3. 9 Pizetti formula for arbitrary signature. - 4 Reconstruction from Line Integrals. - 4. 1 Pencils of lines and John s equation. - 4. 2 Sources at infinity. - 4. 3 Reduction to the Radon transform. - 4. 4 Rays tangent to a surface. - 4. 5 Sources on a proper curve. - 4. 6 Reconstruction from plane integrals of sources. - 4. 7 Line integrals of differential forms. - 4. 8 Exponential ray transform. - 4. 9 Attenuated ray transform. - 4. 10 Inversion formulae. - 4. 11 Range conditions. - 5 Flat Integral Transform. - 5. 1 Reconstruction problem. - 5. 2 Odd-dimensional subspaces. - 5. 3 Even dimension. - 5. 4 Range of the flat transform. - 5. 5 Duality for the Funk transform. - 5. 6 Duality in Euclidean space. - 6 Incomplete Data Problems. - 6. 1 Completeness condition. - 6. 2 Radon transform of Gabor functions. - 6. 3 Reconstruction from limited angle data. - 6. 4 Exterior problem. - 6. 5 The parametrix method. - 7 Spherical Transform and Inversion. - 7. 1 Problems. - 7. 2 Arc integrals in the plane. - 7. 3 Hemispherical integralsin space. - 7. 4 Incomplete data. - 7. 5 Spheres centred on a sphere. - 7. 6 Spheres tangent to a manifold. - 7. 7 Characteristic Cauchy problem. - 7. 8 Fundamental solution for the adjoint operator. - 8 Algebraic Integral Transform. - 8. 1 Problems. - 8. 2 Special cases. - 8. 3 Multiplicative differential equations. - 8. 4 Funk transform of Leray forms. - 8. 5 Differential equations for hypersurface integrals. - 8. 6 Howard s equations. - 8. 7 Range of differential operators. - 8. 8 Decreasing solutions of Maxwell s system. - 8. 9 Symmetric differential forms. - 9 Notes. - Notes to Chapter 1. - Notes to Chapter 2. - Notes to Chapter 3. - Notes to Chapter 4. - Notes to Chapter 5. - Notes to Chapter 6. - Notes to Chapter 7. - Notes to Chapter 8.